Typically, mathematical and engineering systems use a decimal system and/or a binary system to represent numbers and to perform calculations. Over time, alternative numeration systems have been developed for different applications. Such numeration systems include fixed-radix, mixed-radix and mixed-base. A fixed-radix numeration system has a constant radix for all positions of a sequence of digits. In fixed-radix numeration systems, the weights of successive positions are successive integral powers of a single radix, multiplied by the same factor. A mixed-radix numeration system is a radix numeration system in which all radices of each position of a sequence of digits are constant, but not necessarily the same. The mixed-radix numeration system is a more general numeration system in which there may not be integral ratios between the radices of all digits. In a mixed-based numeration system, numbers are represented as the sum of a sequence of position values. Each position consists of a mantissa and a base. The base of a given position is constant for a given application, but the bases across positions are not necessarily integral ratios between the radices of all the positions.
Although the above numeration systems may be used to encrypt data (e.g., prior to transmission over a communication channel), each numeration system has certain disadvantages with respect to transmission efficiency. In addition, encryption systems based on each of the above numeration systems has certain disadvantages with respect to susceptibility to unauthorized deciphering.